Spectral element modelling of fault-plane reßections...

Geophys. J. Int. (2007) 170, 933–951 doi: 10.1111/j.1365-246X.2007.03437.x

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Spectral element modelling of fault-plane reflections arising fromfluid pressure distributions

Matthew Haney,1,! Roel Snieder,1 Jean-Paul Ampuero2 and Ronny Hofmann3,†1Center for Wave Phenomena, Geophysics Department, Colorado School of Mines, Golden, CO 80401, USA. E-mail: [email protected] of Geophysics, Seismology, and Geodynamics, ETH Honggerberg (HPP), Zurich, CH-8093 Switzerland3Center for Rock Abuse, Geophysics Department, Colorado School of Mines, Golden, CO 80401, USA

Accepted 2007 March 9. Received 2007 February 26; in original form 2006 June 15

S U M M A R YThe presence of fault-plane reflections in seismic images, besides indicating the locations offaults, offers a possible source of information on the properties of these poorly understoodzones. To better understand the physical mechanism giving rise to fault-plane reflections incompacting sedimentary basins, we numerically model the full elastic wavefield via the spectralelement method (SEM) for several different fault models. Using well log data from the SouthEugene Island field, offshore Louisiana, we derive empirical relationships between the elasticparameters (e.g. P-wave velocity and density) and the effective–stress along both normalcompaction and unloading paths. These empirical relationships guide the numerical modellingand allow the investigation of how differences in fluid pressure modify the elastic wavefield.We choose to simulate the elastic wave equation via SEM since irregular model geometriescan be accommodated and slip boundary conditions at an interface, such as a fault or fracture,are implemented naturally. The method we employ for including a slip interface retains thedesirable qualities of SEM in that it is explicit in time and, therefore, does not require theinversion of a large matrix.

We perform a complete numerical study by forward modelling seismic shot gathers over afaulted earth model using SEM followed by seismic processing of the simulated data. Withthis procedure, we construct post-stack time-migrated images of the kind that are routinelyinterpreted in the seismic exploration industry. We dip filter the seismic images to highlightthe fault-plane reflections prior to making amplitude maps along the fault plane. With theseamplitude maps, we compare the reflectivity from the different fault models to diagnose whichphysical mechanism contributes most to observed fault reflectivity. To lend physical meaningto the properties of a locally weak fault zone characterized as a slip interface, we proposean equivalent-layer model under the assumption of weak scattering. This allows us to use theempirical relationships between density, velocity and effective stress from the South EugeneIsland field to relate a slip interface to an amount of excess pore-pressure in a fault zone.

Key words: fault zones, fluid pressures, spectral element method.

1 I N T RO D U C T I O N

Seismic data acquisition and processing have evolved to the pointthat fault-plane reflections are often imaged under favorable con-ditions (Liner 1999), such as above salt in the Gulf of Mexico.Reflections originating from fault zones hold important informa-tion about fluid movement along faults (Haney et al. 2005b) and the

!Now at: USGS, Alaska Volcano Observatory, Anchorage, AK 99508, USA.†Now at: Shell International Exploration and Production, Houston, TX77001, USA.

capacity of a fault to act as a seal (Haney et al. 2004). Faults posea challenge to seismic interpreters by virtue of their dual functionas both hydrocarbon traps and pathways (Hooper 1991) for hydro-carbons to move from deep kitchens into shallower, economicallyproducible reservoirs. Any light that seismic data can shed on thissituation would be useful.

To gain a stronger grasp on the factors at play in causing fault-plane reflectivity, we have pursued a complete numerical study ofseismic wave interaction with fault models. By complete, we donot simply model the entire measured (elastic) wavefield with highfidelity, but additionally process the data into time-migrated images,which is how many geoscientists in the petroleum industry gainaccess to and begin examining seismic data. We model the wavefield

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934 M. Haney et al.

with a numerical code based on the spectral element method (SEM)that allows for discontinuous slip to occur at fault planes (Ampuero2002). Processing of the elastic wavefield output by the SEM codehas been accomplished using Seismic Un*x (Stockwell 1997).

Previous studies that examined fault-plane reflectivity have re-lied on simpler or less advantageous numerical methods for forwardmodelling the seismic wavefield than SEM. For instance, both Jones& Nur (1984) and Moore et al. (1995) employed simple 1-D mod-elling techniques based on the convolutional model of the seismictrace (Robinson 1984) to interpret fault-plane reflections in seismicdata. Townsend et al. (1998) performed 2-D finite-difference mod-elling of the seismic wavefield reflected from layers offset by fault-ing, but did not study the reflections from the fault plane itself. Zhu &Snieder (2002) adapted a 2-D staggered-grid, velocity–stress finite-difference technique to model reflections from a fault or fracture.Bakulin et al. (2004) performed 3-D finite-difference modellingover a horizontal fault zone consisting of a network of inclusions, orfractures. The advantages of SEM over finite-difference techniqueshas been discussed previously by Komatitsch et al. (2002).

We sketch the theory behind SEM and, after discussing the dip-filtering step we employ to highlight the fault-plane reflections inmigrated data, we present results for several different fault models.These models represent examples and combinations of three basictypes of heterogeneity we expect to exist at faults. These three basictypes are as follows.

(i) Juxtaposition (sand/shale or shale/sand) contacts.(ii) Pressure contrasts (!P) across the fault.(iii) Locally weak fault zones, that is, slip interfaces.

We expect from the outset that these various types of heterogeneityshow up differently in dip-filtered seismic images. For instance,since the juxtaposition contacts exist over the length scale of a typicalbed thickness and have positive (sand/shale) or negative (shale/sand)reflection coefficients, the smoothing of the dip-filter (Oppenheim& Schafer 1975) should act to suppress this contribution to thedip-filtered fault-plane reflectivity compared to the other two faultmodels. This is desirable since the juxtaposition contacts do notcarry direct information on the sealing or conducting properties ofthe fault.

The other two types of heterogeneity, pressure contrasts and slipat the fault plane, relate to pore-pressure distributions at the fault andare not attacked by the dip filter in the same way as are the juxtaposi-tion contacts. Although reflections due to !P are interpreted in theoil industry more commonly than reflections due to a locally weakfault zone, Worthington & Hudson (2000) have recently explainedthe apparent attenuation of seismic waves transmitted through a faultin the North Sea by allowing slip planes to exist at the fault. In dis-cussing possible causes of interfacial slip, Nihei et al. (1994) statedthat in ‘oil and gas reservoir environments and in parts of the earth’scrust where effective stresses may be low due to the presence ofhigh pore-pressures, the contact between neighbouring lithologiesor across fractures may be imperfect.’ Given this, we expect that alocally pressurized fault zone can act as a slip interface and giverise to reflected seismic waves. Lending importance to the modelof a pressurized fault zone, there have recently been reports of fieldobservations of elevated pore-pressure inside fault zones (Crampinet al. 2002; Haney et al. 2005b).

In the first section of this paper, we discuss empirical relationshipsbetween pore-pressure and three basic rock properties—porosity,density and sonic velocity. The data for this analysis come fromwells drilled at the South Eugene Island field, offshore Louisiana.These relationships form the basis for the models used in the subse-

quent SEM simulations. The fact that pore-pressure largely controlsrock matrix properties in compacting sedimentary basins allowsmethods for imaging seismic reflections to indirectly measure spa-tially varying pore-pressure distributions. The variation of the threerock properties with effective stress reveals a fundamental hysteretictype of behaviour in the sediments. Evidence for both plastic (irre-versible) and elastic (reversible) deformation exists in the availablewell data and pressure tests. These two regimes point to differentunderlying causes of overpressure (Hart et al. 1995). For these dualdeformation mechanisms, we construct two empirical relationshipsbetween each rock property and pore-pressure—one valid for eachregime. Before we discuss these issues, though, we wish to clarifyexactly what we mean by the term effective stress in the rest of thispaper.

2 V E RT I C A L E F F E C T I V E S T R E S S

Pore-pressures that exceed the hydrostatic pressure, or overpres-sures, lead to a lowering of density and seismic velocity andmay contribute to the reflectivity of associated with fault zones.Pennebaker (1968) was among the first geoscientists to demonstratethe ability of seismic stacking velocities to detect fluid pressures inthe subsurface. Terzaghi (1943), however, had previously discussedthe basic principle, that of an effective stress acting on the rockframe. According to Terzaghi’s principle, the effective stress deter-mines rock properties (e.g. P-wave velocity). In extensional regimestypical of sedimentary basins, it turns out that, to a good approx-imation, only the vertical effective stress needs to be consideredsince the sediments compact by uniaxial strain (Engelder 1993).Terzaghi’s definition of the vertical effective stress, now known asthe differential stress " d (Hofmann et al. 2005), is the differencebetween the vertical confining stress, " v , and the pore-pressure p:

"d = "v # p. (1)

Eq. (1) states that rocks of similar composition but at different con-fining stress and pore-pressure have the same velocity so long asthe difference between the confining stress and pore-pressure is thesame. Hence, high pore-pressure, which lowers effective stress, leadsto lower seismic velocities.

Following the work of Terzaghi, rock physicists began to questionwhether the effective stress governing rock properties is simply thedifference between the confining stress and the pore-pressure (Wang2000; Hofmann et al. 2005). Today, the most general effective–stresslaw is instead

"e = "v # np, (2)

where the parameter n is called the effective stress coefficient.Carcione & Tinivella (2001) state that the value of n can differ foreach physical quantity (e.g. permeability, compressibility, or shearmodulus), and that it depends linearly on the differential stress ofeq. (1). Currently, the effective–stress coefficient is a controversialtopic and is an active area of research within the rock physics com-munity. For the remainder of this paper, we do not distinguish be-tween differential stress, " d and effective stress, " e; that is, we taken = 1 in eq. (2). This assumption is commonly made in the petroleumexploration industry and is supported in arguments posed recentlyby Gurevich (2004). The assumption of n = 1 is also best suited forhigh porosity, poorly consolidated rocks, as are found in compactingsedimentary basins.

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Fluid pressure and fault reflectivity 935

3 P O RO S I T Y V E R S U S D E P T H

Compaction acts to reduce the porosity of sediments as they areburied; however, this process can continue only as long as fluids inthe diminishing pore space are allowed to be expelled. Such wouldbe the case in normally pressured, hydrostatic sediments in whichthe fluids are in communication up to the seafloor. Once the move-ment of the fluids out of the pore space is opposed, as in a compart-ment sealed-off by low permeability or high capillary-entry-pressureshales or fault gouge, the porosity remains constant with burial depthif the fluid is more or less incompressible. This situation is called un-dercompaction (Huffman 2002). Undercompaction means the sedi-ments are ‘frozen’ in time and are simply buried in their unchangingearlier compaction state (Bowers 1995). To compound the situa-tion, if fluid from outside the undercompacted sediments is pumpedinto the pore space, or if hydrocarbons are generated from withinthe undercompacted sediments, a process called unloading occurs(Huffman 2002). Whereas undercompaction can only cease the re-duction of porosity (Bowers 1995), unloading can actually reversethe trend and increase porosity. Although unloading can reverse thetrend, it cannot reclaim all of the previously lost porosity. This isbecause the compaction process has a large irreversible component.In contrast, unloading and loading of sediments by pumping fluidinto and then depressurizing the pore space is a reversible process,insofar as the fluid does not cause hydrofracturing.

We have examined wireline data taken in wells at the South Eu-gene Island field, offshore Louisiana, for indicators of overpressure,such as constant porosity as a function of depth. Previous work byHart et al. (1995) shows the crossover from hydrostatic to over-pressured conditions in porosities derived from sonic velocities. Wetake a slightly different, perhaps more straightforward, approachbased on the density log. The South Eugene Island field is a Plio-Pleistocene minibasin formed by salt withdrawal and has yieldedmore than 300 million barrels of oil in its lifetime. An illustrationof the main subsurface features at South Eugene Island is shown inFig. 1. The main part of the field is a vertical stack of interbeddedsand and shale layers bounded by two large growth faults to thenorth and south.

Fig. 2 shows porosity derived from density logs within the mini-basin taken in the following wells at South Eugene Island: A13,A20ST, A14OH, A15, A23, A6, B10, B1, B2, B7 and B8. Becausethe geology in the minibasin is essentially horizontally layered, weignore the fact that some wells may be far away from each otherand simply look at the depth variation of their porosity. In all thewell logs shown in this paper, we have done some smoothing withdepth (over $100 m) to remove any short-range lithologic influ-ences (e.g. interbedded sand and shale) on the density and velocity.To obtain the porosity from the density log, we take the solid grainsto have a density of 2650 kg m#3 and the fluid to have a densityof 1000 kg m#3, as in Revil & Cathles (2002). In Fig. 2, there is aclear break from the shallow, decreasing porosity trend at a depthof 1800 m. Based on the work of Stump et al. (1998), we assumethat this is the onset of overpressures in the sedimentary section,beneath a shale bed located above a layer called the JD-sand. We fitan exponential trend to the porosity values above 1800 m, known asAthy’s Law (Athy 1930), to get the normal compaction trend in thehydrostatically pressured sediments

#c(z) = 0.47 e#0.00046 z, (3)

where in this equation, the depth z is in metres. The superscript c ineq. (3) refers to the fact that this functional relationship characterizesnormal compaction. In the porosity-versus-depth plot of Fig. 2, this

Figure 1. Regional map of the Gulf of Mexico (top) showing the study area,the South Eugene Island field. The four main faults at the field are shownin the qualitative depiction of a typical depth section (bottom) as the A, B,F and Z faults. Throw across the faults is shown by the single layer runningfrom left to right. Most of the wells at South Eugene Island were drilled intothe shallow, hydrostatic section within the minibasin, which is bounded bythe Z fault to the south and the A- and B-faults to the north. The A20ST wellwas unusual in that it was continued through the A-fault system and intothe deeper, overpressured and upthrown block to the north of the minibasin.There, two pressure measurements (RFTs) were made at the positions shownby 1 and 2 in the depth section.

relationship holds for any movement towards the right on the normalcompaction curve and any purely right-going horizontal deviationsfrom the normal compaction curve. For purely right-going horizon-tal deviations, the depth z used in eq. (3) is equal to the depth at whichthe horizontal deviation started. The two circles in Fig. 2 representsamples taken in the A20ST well in northern upthrown block (seeFig. 1) and are connected to the normal compaction curve by bothhorizontal and vertical dashed lines. The vertical dashed lines showthe departure of the samples from the normal compaction trend. Wereturn to these in the next section.

The sediments deeper than 1800 m in Fig. 2 maintain a nearlyconstant porosity of around 0.2 during subsequent burial (a hori-zontal deviation from the compaction trend). Though the depth ofthe sediments increases with burial, the effective stress experiencedby the sediments does not appear to change and compaction ceases.Hence, the additional weight of the overburden with increasing depthis borne by the fluids trapped in the pore space. As a result, thepore-pressure increases with the vertical gradient of the overburdenstress, in order to satisfy Terzaghi’s law eq. (1), and is said to have


936 M. Haney et al.

0 1000 2000 30000

0.1

0.2

0.3

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0.5

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poro

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Figure 2. Porosity versus depth at South Eugene Island. The thick, solid lineis the best-fitting normal compaction trend using Athy’s Law (Athy 1930).The faint solid lines are density-derived porosity values from 11 wells atSouth Eugene Island. To obtain the porosity, we assume that the solid grainshave a density of 2650 kg m#3 and the fluid has a density of 1000 kgm#3, as in Revil & Cathles (2002). There is a clear break from the shallow,exponentially decreasing porosity trend at a depth of 1800 m, at which pointthe porosity remains constant with increasing depth, as shown by the lowestdashed line. The two circles are density-derived porosities from samples inthe upthrown block to the north of the minibasin at South Eugene Island (seeFig. 1). The two dashed lines connecting the circles to the main compactiontrend are the interpreted porosity histories of the two samples. They show aperiod of undercompaction, depicted as a horizontal dashed line deviatingfrom the normal compaction trend, followed by an unloading path, shown asa vertical dashed line, due to a late-stage pore-pressure increase.

a lithostatic gradient. In doing so, overpressure, or pore-pressure inexcess of hydrostatic, is created below 1800 m.

4 D E N S I T Y V E R S U S V E RT I C A LE F F E C T I V E S T R E S S

Since density is a parameter widely used in the field of seismicwave propagation, we study the variability of the bulk density inthis section. In contrast to the preceding section, we want to seehow density changes with effective stress, not depth. To accomplishthis, we take only the measurements that are shallower than 1800 m,where the pore pressure is, by all indications, hydrostatic. Therefore,we know the pore pressure and can calculate the effective stress. Inoverpressured compartments, since the pore-pressure is unknown,direct measurements by Repeat Formation Tests (RFTs) are neces-sary to calculate the effective stress.

We rewrite eq. (3) in terms of density and effective stress usingthe relationships

$ = $s(1 # #) + #$ f , (4)

and

"d = $ f gz, (5)

where $ is the bulk density and $ s and $ f are the densities ofthe solid and fluid components. Note that the relationship for " d

holds only under hydrostatic conditions. From these relationshipsand eq. (3), we obtain the normal compaction curve for density

$c("d ) = $s # 0.47 ($s # $ f ) e#0.0003"d , (6)

0 1000 2000 3000 40001800

2000

2200

2400

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ity (

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3 )

1

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2

Figure 3. Density versus effective stress at South Eugene Island. The thicksolid line is the same normal compaction trend shown in Fig. 2, excepttransformed into density and effective stress. The faint solid lines are alsothe same as in Fig. 2, except that they are now limited to the hydrostatic depthsdown to 1800 m. The circles represent two pressure measurements, labelled1 and 2, which were made in the overpressured upthrown block where adensity log also existed. For each pressure measurement, we plot the datapoint twice—one where it should lie on the normal compaction curve wereit to have been normally pressured, and the other where it actually does plotbecause of severe overpressure. Note that sample 1 is from a greater depththan sample 2.

where $ s and $ f are the densities of the solid and fluid compo-nents, taken as 2650 and 1000 kg m#3, respectively, and " d is inpsi. We plot this normal compaction curve in Fig. 3 together withthe density measurements. Also, in Fig. 3, we show as circles twodata points obtained from RFT pressure measurements and densitylog measurements in the overpressured upthrown block. We showthe circles in two locations—one on the normal compaction trendwhere they would plot if the measurements were at hydrostaticallypressured locations, and the other where they actually plot becauseof severe overpressures being present in the upthrown block.

At this point, we don’t know exactly how the samples taken in theupthrown block came to be off the normal compaction trend. Usinga laboratory measurement of the unloading coefficient by Elliott(1999) on a core sample taken near the locations of samples 1 and2, the path that these samples took to their present locations can beestimated. Elliott (1999) characterized the effect of unloading, orelastic swelling on the porosity of the core samples to be

#u("d ) = #0 (1 # %"d ) , (7)

where #0 and % characterize the deviation of the unloading path fromthe normal compaction trend. Note the superscript u, in contrast toeq. (3), indicating the unloading path instead of the normal com-paction trend. Elliott (1999) found that #0 = 0.37 and % = 0.98 %10#8 Pa#1 for the unloading path. Though these parameters de-scribe the porosity, we use them to find the slope of the unloadingpath for density using the relationships between porosity and den-sity described earlier. After finding this slope, we can construct theunloading path for the density using eq. (6)

$u("d ) = 0.04 ("d # "max) + $s

#0.47 ($s # $ f ) e#0.0003"max . (8)

This expression contains an extra parameter " max that refers to thevalue of the effective stress when the sample began to be unloaded,



or the maximum past effective stress. We do not know " max forsamples 1 and 2, but we do know that " max must lie on the maincompaction trend. Hence, we can construct linear unloading pathsfor the density, as shown by the dashed lines in Fig. 3. With theseunloading paths, we can then find the value for the maximum pasteffective stress " max. It is worth mentioning that the maximum pasteffective stress for sample 1 comes out to be $1500 psi by ourapproach of using Elliott’s experimental results. In an independentmeasurement, Stump & Flemings (2002) performed uniaxial straintests on a core sample taken from the same location as sample 1 tofind the maximum past effective stress. Stump & Flemings (2002)report a value of 1248 psi for this sample, close to our estimate of$1500 psi; visually, the discrepancy lies within the error bars of thenormal compaction curve’s fit to the density log data.

With the estimate of the maximum past effective stress, we canalso return to Fig. 2 and find the depth at which samples 1 and 2left the normal compaction trend, since in the hydrostatic zone thedepth is a linearly scaled version of the effective stress. These depthscorrespond to a slightly lower porosity than that of samples 1 and 2.We interpret this as being the result of a late stage porosity increaseand represent it as unloading paths, shown by vertical dashed lines,for samples 1 and 2 in Fig. 2.

5 S O N I C V E L O C I T Y V E R S U SV E RT I C A L E F F E C T I V E S T R E S S

For the purposes of modelling faults and to make inferences aboutthe distribution of pore-pressure from seismic interval velocity in-versions, accurate pore-pressure-versus-velocity relationships arecritical (Dutta 1997). In general, sonic velocity has a normal com-paction curve and unloading paths as a function of effective stressthat are similar to those we just described for the density well logdata. To obtain these relationships for velocity, we proceed as forthe density logs: (1) we take 12 shallow wells to make up a dataset of sonic velocity versus effective stress; (2) we select the depthrange with hydrostatic pressures and plot the sonic velocity versuseffective stress; (3) we fit this with a power-law relation for the nor-mal compaction trend and (4) we look at where the two samplesfrom the overpressured upthrown block lie and construct unloadingcurves using the estimate for the maximum past effective stress weobtained in the previous section. The wells we use for characterizingthe sonic velocity come from A20ST, A14OH, A23, A6, B10, B1,B2, B7, B8, A1, B14 and B20.

In Fig. 4, we plot the normal compaction trend for sonic velocityas a thick solid line described by the power-law equation (Bowers1995)

vcp("d ) = 1500 + 2.3 " 0.77

d , (9)

where v p is in m s#1 and " d is in psi. Note again the superscript cfor the normal compaction relation. We also construct the unloadingcurve for v p following the relationship first suggested by Bowers(1995)

vup("d ) = 1500 + 2.3

!

"max

"

"d

"max

#1/6.2$0.77

, (10)

where " d and " max are in psi and v p is again in m s#1.To model elastic waves, one other parameter is needed in addition

to $ and v p; for instance, a seismologist would naturally want theshear velocity. In the absence of information on the shear wavevelocity v s and pressure in the shallow, hydrostatic sediments, we

0 1000 2000 3000 40001500

2000

2500

3000

!v – P (psi)

soni

c ve

loci

ty (

m/s

)

2

1

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Figure 4. Sonic velocity versus effective stress at South Eugene Island.The thick solid line represents the normal compaction curve fitted to theshallow well data, shown in the faint solid lines. We also plot samples 1and 2 both where they should fall on the normal compaction trend, werethey to be normally pressured, and where they actually plot due to the severeoverpressure where they were obtained. Using the estimate for past maximumeffective stress from the density plot and the Bowers-type relation (Bowers1995) shown in eq. (10), we are able to construct the velocity unloadingcurves, shown as dashed lines.

assume that

vs("d ) = vp("d ) # 1500, (11)

where the velocities are in m s#1 and the relationship holds onboth the normal compaction curve and unloading paths. The datapresented by Zimmer et al. (2002) for unconsolidated sands supportsthis assumption, in that the dependence they found for v s on effectivestress is essentially a down-shifted version of the v p curve. Anadditional piece of supporting evidence comes from the only v s dataavailable at South Eugene Island, a shear log from the A20ST well,where samples 1 and 2 were taken (Anderson et al. 1994). There, theratio of v p/v s from the sonic and shear logs falls between 3 and 3.5in the overpressured upthrown block. Inserting the values for v p atsamples 1 and 2 into eq. (11) to get v s and finding the correspondingratio of v p/v s , we get v p/v s $ 3.5 at sample 1 and v p/v s $ 3.0 atsample 2, within the range of the ratios observed in the sonic andshear logs.

To summarize, we have established two empirical relationshipsbetween each of three basic rock properties and pore-pressure atthe South Eugene Island field. Most importantly for subsequentnumerical modelling of wave propagation, we have found relation-ships for the density $ and the sonic velocity v p on both the nor-mal compaction and unloading paths. Without shallow informa-tion on the shear velocity v s , we must make the assumption thatit is a down-shifted version of the v p(" d ) relationship. From look-ing at the density-derived porosity-versus-depth-relationship, we areable to conclude that the deep, overpressured sediments below theJD-sand are predominately overpressured because of undercom-paction, since their porosity did not change appreciably with depth.In contrast, both undercompaction and unloading have contributedto the current overpressured state of the sediments on the upthrownside of the minibasin. We assume that the sediments within theminibasin bounding growth fault zones (faults A and B in Fig. 1)are in a similar compaction state as the upthrown block since Loshet al. (1999) have stated that these ‘fault zones are typically at the


938 M. Haney et al.

same pressure as the upthrown sediments.’ Our conclusion that theupthrown sediments have undergone both undercompaction and un-loading is in agreement with a previous study by Hart et al. (1995) onporosity and pressure at South Eugene Island. We use the above em-pirical relationships between the elastic parameters and fluid pres-sure in the following sections to simulate fault-plane reflections fromdifferent pressure distributions in the subsurface.

6 T H E S P E C T R A L E L E M E N T M E T H O D

Numerical modelling of wave propagation in the Earth can be basedon the weak (Zienkiewicz & Taylor 2000) or strong forms (Boore1970) of the elastodynamic equations of motion. By weak andstrong, we mean the integrated or differential forms of the equa-tions of motion. SEM is based on the weak form and naturallyhandles general geometries and exotic boundary conditions. In thefinite-difference method (based on the strong form), it is notori-ously difficult to implement a linear-slip boundary condition (Coates& Schoenberg 1995) or any general boundary condition for thatmatter (Boore 1970; Kelly et al. 1976). On the other hand, SEMwith explicit time-stepping does not require the inversion of a largematrix, a property usually identified with finite-difference methodsand lumped-matrix finite-element methods. Formally, this propertystems from the diagonality of the mass matrix, a feature that isconsistently obtained by design of the SEM through a combinationof subintegration and the choice of coincident interpolation andquadrature nodes. Although this design is in contrast to the classi-cal and artificial mass-lumping in FEM, the SEM mass matrix canalso be obtained from lumping of the exactly integrated mass matrix(Karniadakis & Sherwin 1999). SEM has the additional property ofspectral convergence, meaning that, as the polynomial order of thebasis functions is increased, the numerical error goes down expo-nentially (Karniadakis & Sherwin 1999). The practical implicationof spectral convergence is the low number of nodes per wavelengthrequired to reach a given accuracy, as demonstrated by the dispersionanalysis of Thompson & Pinksy (1994).

The term ‘spectral element’ indicates that SEM is a mixture offinite-element and spectral methods (Komatitsch & Vilotte 1998).As a result, there are two parameters relevant to the mesh in SEM:the size of the elements and polynomial degree (n # 1, where n is thenumber of zero crossings of the basis functions used within each el-ement). Komatitsch & Tromp (1999) refer to these parameters whenthey speak of the global mesh and the local mesh. Concerning thelocal mesh, there is a known trade-off between accuracy and numer-ical cost (Seriani & Priolo 1994), which suggests that polynomial

degrees no higher than 10 should be used within the elements. Forthe numerical examples in this paper, we use a polynomial degreeof eight, n& = 8.

7 M O D E L L I N G FAU LT E D S T RU C T U R E S

The flexibility provided by SEM makes possible the simulation ofseismic data for several fault models. In addition to numericallymodelling the full 2-D elastic wavefield, we process the SEM mod-elled data into its migrated image. Thus, our procedure representsa complete modelling and processing sequence. Fig. 5 depicts thegeometry of the basic faulted structure we study. The faulted struc-ture contains several layers and a slip interface confined to the faultplane (see Fig. 5). Different models, discussed shortly, share thissame faulted structure but differ in the values of the material proper-ties assigned to the various layers and the slip interface. The faultedstructure depicts a normal fault with a vertical throw of 20 m, avalue characteristic of a small fault. The normal fault dips at 45&.The faulted structure shown in Fig. 5 is similar to the one previouslystudied by Townsend et al. (1998) to assess changes in seismic at-tributes caused by faults disrupting the lateral continuity of events.We discuss the material properties used for the different modelsshortly.

We mesh the interior of the fault structure shown in Fig. 5using a freely available mesh program developed by INRIA,called EMC2. The program can be downloaded at: http://www-rocq.inria.fr/gamma/cdrom/www/emc2/eng.htm. As discussed inthe previous section, SEM has both a local and global mesh. Theglobal mesh is interactively built first using the EMC2 program.Once the global mesh is built, the local mesh is computed auto-matically within the SEM code. For the examples in this paper, weuse a semi-structured global mesh since the faulted structure shownin Fig. 5 is not overly complex. A semi-structured global mesh isdesirable, as opposed to an unstructured mesh, since the accuracyof SEM depends on the Jacobian of the transformation between agenerally shaped element and a standard rectangular element overwhich the integration is performed. Although the global mesh we usehas structure, it honoors the slanted boundaries of the fault. Notethat this would not be possible using a rectilinear ‘checkerboard’grid as in finite-difference methods. In that case, the slanted faceof the fault would be represented by a ‘stair-stepping’ pattern. Thistype of model would give rise to artificial diffractions from the faultplane. After initial construction of the global mesh for SEM, thequadrangle elements comprising the mesh are regularized so thattheir shapes mimic rectangles as closely as possible. For the faulted

Figure 5. The numerical model (left-hand panel) with a zoom-in of the normal fault (right-hand panel). The zoom area is shown on the numerical model with adashed rectangle. A seismic survey has been simulated numerically using this model; details of the survey parameters are given in the text. In the zoom, layersare labelled with numbers 1–12 corresponding to the material properties for models listed in subsequent tables. For models with a slip interface at the fault, theportion of the fault plane that slips is shown by a thicker line in the zoom.



structure in Fig. 5, the entire 2 km % 4 km model is made up of 80 %160 global mesh elements. Thus, the global mesh is made up ofquadrangles that are approximately rectangles of dimension 25 m %25 m. Each global mesh element is further broken down into64 (= n2

&) smaller local elements as defined by the employed poly-nomial degree (n& = 8), as discussed in the previous section.

We simulate a seismic survey over the faulted structure in Fig. 5using 11 seismic sources. The first source is located 1 km fromthe leftmost edge of the model and the source array continues for2 km at a spacing of 200 m (or a 200 m shot interval). In additionto the sources, there are 241 receivers employed in the simulatedseismic survey. The first receiver is located 500 m from the leftmostedge of the model and the receiver array continues for 3 km at aspacing of 12.5 m. The receiver array is kept constant when excitingthe different sources. We forward model the elastic wavefield byrunning the SEM code in serial (one compute node for each shot)on a 32-processor pentium IV Xeon (3.0-GHz) cluster. All of thesubsequent processing of the modelled wavefield is performed on aworkstation using the Seismic Un*x package (Stockwell 1997).

Tables 1 and 2 show the compressional velocity and the densityof four models for the faulted structure shown in Fig. 5. These fourmodels represent four different states of overpressure in the foot-wall of the faulted structure. Assuming a hydrostatic pressure of2800 psi at the depth of the fault, the different overpressure states

Table 1. P-wave velocities for models 1–4 used in the SEM simulations. Thevertical throw between the upthrown (layers 2–6) and downthrown (layers7–11) sediments is 20 m. The geometry of the faulted structure is given inFig. 5. The S-wave velocity for each layer is 1500 m s#1 less than the P-wavevelocity, as shown in eq. (11).

Layer Thickness v p (m s#1) v p (m s#1) v p (m s#1) v p (m s#1)(m) Model 1 Model 2 Model 3 Model 4

1 900 2600 2600 2600 26002 50 2750 2570 2660 27053 30 2600 2380 2490 25454 50 2750 2570 2660 27055 30 2600 2380 2490 25456 90 2750 2570 2660 27057 50 2750 2750 2750 27508 30 2600 2600 2600 26009 50 2750 2750 2750 275010 30 2600 2600 2600 260011 90 2750 2750 2750 275012 850 2600 2600 2600 2600

Table 2. Densities for models 1–4 used in the SEM simulations. The verticalthrow between the upthrown (layers 2–6) and downthrown (layers 7–11)sediments is 20 m. The geometry of the faulted structure is given in Fig. 5.

Layer Thickness $ (kg m#3) $ (kg m#3) $ (kg m#3) $ (kg m#3)(m) Model 1 Model 2 Model 3 Model 4

1 900 2240 2240 2240 22402 50 2280 2240 2260 22703 30 2240 2210 2225 22324 50 2280 2240 2260 22705 30 2240 2210 2225 22326 90 2280 2240 2260 22707 50 2280 2280 2280 22808 30 2240 2240 2240 22409 50 2280 2280 2280 228010 30 2240 2240 2240 224011 90 2280 2280 2280 228012 850 2240 2240 2240 2240

yield pore-pressure contrasts !P at the fault of 0 psi (Model 1), 600psi (Model 2), 300 psi (Model 3) and 150 psi (Model 4). From themodel with no pore-pressure contrast (Model 1), it can be seen fromTables 1 and 2 that there are two basic rock types: an acousticallyhard shale ($ = 2280 kg m#3, v p = 2750 m s#1 and v s = 1250 m s#1)in layers 2, 4, 6, 7, 9 and 11 and an acoustically soft sand ($ =2240 kg m#3, v p = 2600 m s#1 and v s = 1100 m s#1) in layers 3,5, 8 and 10. The overburden and underburden are given the prop-erties of the sand. The properties of the sand in Model 1 are takenfrom a well log (in the A20ST well) that intersected a sand layerat the South Eugene Island field known as the JD-sand. The shalevalues in Model 1 come from the lower bounding shale beneaththe JD-sand. The properties of the other three models (2, 3 and 4)are calculated using the pore-pressure relations described earlier forthe case of pure undercompaction. Thus, the pore-pressure relationsgive the numerical modelling a physical basis applicable to the SouthEugene Island field. Note that the model with no pore-pressure con-trast (Model 1) simply has juxtaposition contrasts across the fault.Thus, by comparing the reflectivity of the four models with differ-ent pore-pressure contrasts, we are able to compare reflectivity dueto juxtapositions to reflectivity dominated by a strong !P at thefault plane. This is relevant to the observed fault-plane reflectivityat South Eugene Island since pressure measurements taken near thelarge, minibasin-bounding growth fault, known as the A fault, showa 780 psi increase in pore-pressure over 18 m in going from the hy-drostatically pressured downthrown sediments to the overpressuredupthrown sediments (Losh et al. 1999). A fault with such a large andsharp !P, compared to the dominant wavelength of seismic waves($100 m), should reflect the seismic waves due to both the juxtapo-sitions and the !P across the fault. By comparing the reflectivity ofthe four different models, we should be able to estimate how smalla pore-pressure contrast can be and still be seismically detectable.

We utilize the advantages of SEM modelling, as described ear-lier, to accommodate interfacial slip at the fault in addition to thejuxtaposition contacts and !P across the fault. We chose to imple-ment slip interfaces with a linear slip law: further detail about theseslip interfaces is given in Appendix A. In this study, we examinefour different values of slip at an interface, as shown in Table 3.The slip interfaces are fully specified in terms of their normal 'N

and tangential 'T (or shear) compliances. We adopt a relation be-tween the two compliances given by 2'N = 'T , following Chaisri& Krebes (2000). This relation between the two compliances mayalso be found in the experimental data of Pyrak-Nolte et al. (1990).Specifically, in Table 4 of Pyrak-Nolte et al. (1990), the relation2'N = 'T is given for sample E30 at a confining stress of 20 MPaand under dry conditions. We refer to the four slip models in orderfrom the most to the least slipping as Slip Model A, B, C and D.Since these slip models only concern the boundary condition at thefault plane, we may insert these different slip models into the pore-pressure models shown in Tables 1 and 2. As an example, we mayspeak of Slip Model A embedded in Model 1 (the model with nopore-pressure contrast across the fault) and so on. The four mod-els of pore-pressure contrasts (Model 1 through 4) and four modelsof slip interfaces (Slip Models A–D) thus provide many possiblecombinations for this SEM modelling study. Note that, for the slipinterfaces, the entire fault-plane does not slip—only a portion of itas shown in Fig. 5.

The values for the normal and tangential compliances given inTable 3 are, in general, orders of magnitude larger than those ob-served in laboratory or field data (Worthington 2006; Worthing-ton & Lubbe 2007). We have selected these values for the com-pliances so that in our numerical simulations reflections from


940 M. Haney et al.

Table 3. Four different slip interfaces described in terms of their normal and tangential compliances.These different slip interfaces are used in the SEM modelling.

Slip-Model Normal compliance, 'N (m Pa#1) Tangential compliance, 'T (m Pa#1)

A 5.0 % 10#10 1.0 % 10#9

B 3.5 % 10#10 7.0 % 10#10

C 2.5 % 10#10 5.0 % 10#10

D 1.0 % 10#10 2.0 % 10#10

Table 4. Different slip interfaces described in terms of their effective layerparameters assuming a thickness of 10 m and various maximum verticaleffective stress, " max. The estimates are made under the assumption thatthe fault rock began its unloading path after reaching its maximum verticaleffective stress. Relationships between velocity and pore-pressure derivedearlier in the paper are used to relate the compliances of the different slipinterfaces to vertical effective stress " e and pore-pressure p at South EugeneIsland. Also the pore-pressure estimate assumes a depth of 1850 m, wherethe overburden stress is 5500 psi and the hydrostatic pressure is 2800 psi.

Slip-Model " max $ (kg m#3) vp (m s#1) " e (psi) p (psi)

A 2800 2200 2020 10 5490B 2800 2210 2180 90 5410C 2800 2220 2290 320 5180D 2800 2270 2470 1660 3850A 2400 2180 2040 30 5470B 2400 2190 2190 240 5260C 2400 2210 2300 780 4730A 2000 2150 2060 110 5390B 2000 2170 2200 700 4800C 2000 2220 2290 1830 3670A 1600 2120 2080 450 5050

individual slip interfaces are strong enough to show up distinctly.Vlastos et al. (2003) adopt a similar approach in their numericalmodelling study of fractures. Such large compliances could easilybe explained by noting that, for instance, the excess normal compli-ance is given by the fracture density multiplied by the normal com-pliance of a single fracture (Worthington & Lubbe 2007). For highfracture densities, as are typically observed at fault zones, the com-pliances given in Table 3 could, therefore, be realizable since theywould represent a high fracture density multiplied by a much smaller(and physically possible) single fracture compliance. As pointed outby Worthington & Hudson (2000) though, such high fracture den-sities at fault zones are typically complex and are probably not welldescribed by sets of fractures which are aligned and larger than aFresnel zone. Although the compliances given in Table 3 are un-physically large, as discussed in a following section on the resultsof our SEM-based numerical simulations, it turns out that we canestimate the responses for smaller values of the compliances sincethe slip interfaces given in Table 3 are in the weak scattering regime.In the weak scattering regime, the strength of the reflection is, to ahigh degree of approximation, linearly proportional to the normaland tangential compliances, as described in Appendix A. Thus, in afollowing section, we linearly extrapolate our numerical results andestimate the minimal values of the compliances that give a notablereflection.

Although the values for the compliances listed in Table 3 are un-physically large, in a later section of this paper we present a weakscattering model to relate the reflections from the slip interfaces toreflections from thin layers. These thin layers represent a simplemodel of a finite thickness fault zone. Thus, the reflection from aparticular slip interface is equivalent to the reflection from a ‘family’of thin layers of varying thicknesses and material properties. Given

the properties of these thin layers, we can use the pore-pressure re-lationships derived earlier to investigate the values of pore-pressurein a finite thickness fault zone which reflects waves equivalently toa slip interface. For now, the values for the compliances listed inTable 3 are simply parameters describing the degree of weldednessbetween the surfaces on either side of the slip interface.

As with many numerical methods, construction of the global meshis tied to the material properties (specifically the P- and S-wavevelocities) of the different fault models. For example, it is oftenquoted that there must be at least 5 gridpoints per minimum desiredwavelength for SEM modelling to be accurate and not be corruptedwith numerical dispersion (Komatitsch & Tromp 1999). This can beexpressed as (min/(h/n&) $ 5. Note that in this criterion the lengthof the global mesh element h is divided by the polynomial orderused for the local mesh n&. This approximation assumes the localmesh elements are equally spaced within the global mesh element(Komatitsch & Tromp 1999). Since we initiate our simulations withan explosive source whose time function is a Ricker wavelet witha dominant frequency f 0 of 20 Hz, we may take the maximumdesired frequency to be 50 Hz, such that f max = 2.5 f 0. Since theSEM simulations we perform are elastic, the minimum wavelength(min is determined by the minimum S-wave velocity. The minimumS-wave velocity over all of the fault models is, from Table 1, givenby 880 m s#1 and the minimum wavelength (min is determinedby the minimum S-wave velocity divided by the maximum desiredfrequency. Thus, (min = 17.6 m. As stated before, a typical globalmesh element has sides of length h = 25 m and the polynomialdegree employed in the simulations is n& = 8. Therefore, at leastabout 5.6 gridpoints exist per minimum desired wavelength for allof the models.

In addition to the need for 5 gridpoints per minimum desiredwavelength, there are also numerical stability considerations. In con-trast to numerical dispersion considered above, numerical stabilityis determined by the maximum propagation velocity, which for theelastic SEM code is the maximum P-wave velocity of 2750 m s#1

(see Table 1). Stability is formally expressed by the CFL-type cri-terion

vmax!thmin

' 0.5, (12)

where !t is the time step and hmin is the minimum length of a localmesh element in the model. Again using the approximation that aglobal mesh element, with a side of length 25 m broken up into n&

local mesh elements, hmin = 25 m/8 = 3.125 m for n& = 8. Alongwith the fact that vmax is equal to 2750 m s#1, this gives the criteriafor a maximum time step to ensure stability as !t ' 0.00057 s.We chose to execute the SEM examples shown in this paper with!t = 0.0001 s to ensure stability. We include a safety factor ofapproximately 5 in our choice of !t since the value for hmin weuse is a high estimate: the local mesh elements for SEM are not infact equally spaced within a global mesh element (Komatitsch &Tromp 1999). This unequal spacing leads hmin to be smaller thanour earlier estimate. After executing the SEM code with this value



of !t , we simply down-sample the seismograms by taking every20th sample to simulate seismic sections with a sampling rate of2 ms, as is common in the seismic industry.

Since the SEM code is elastic, both primary PP-reflections andconverted PS-reflections show up on the vertical component of thedisplacement seismograms. We mute the converted waves in orderto proceed with conventional P wave time-processing. We subtractoff the direct waves (P, S and Rayleigh) by running a homogeneoussubsurface simulation with the elastic properties of the overbur-den (shown as layer 1 in Fig. 5). After this step, we perform ageometrical-spreading correction to compensate for the (approxi-mately) spherical divergence of the wavefront emanating from thesource, a normal moveout (NMO) correction to flatten reflectionsfrom horizontal layers in a common-midpoint gather (CMP gather),a dip moveout (DMO) correction to flatten reflections from dippingreflectors (e.g. the fault-plane reflection), and finally a summationover different source-receiver offsets for a single CMP (a stack) tosimulate zero-offset data (Yilmaz 1987; Black et al. 1993). With thesimulated zero-offset section, we proceed with a constant-velocitymigration using the velocity of the overburden. A source of errorin this simulation originates from the slight undermigration of thedeepest reflectors and the fault-plane reflection. We chose to migratewith constant velocity since we have interpreted time-migrated seis-mic sections in the Gulf of Mexico (Haney et al. 2004, 2005b) andwanted the modelling and processing sequence to mimic the ob-served data as closely as possible.

There is one final processing step applied to the migrated data.After producing seismic images, we apply a dip filter in the )–kdomain to highlight the fault-plane reflections relative to the reflec-tions from horizontal layers. We use the following transfer functionfor the dip filter

K (), k) = 12n + 1

%

2 sin[(n + 1)()pst + k)h/2]sin[()pst + k)h/2]

% cos[n()pst + k)h/2] # 1&

. (13)

This is the form of a dip filter that corresponds to stacking 2n +1 traces centred about an output point along a dip pst. The param-eter h is the distance between the traces (assumed constant). Analternative procedure would be a combination of interpolation andslant stacking in the t–x domain; however, the )–k dip filter is suf-ficiently accurate for the examples shown here. Fig. 6 depicts thesimulated reflection images for the two of the fault models next totheir dip-filtered versions which highlight the fault-plane reflection.The dip filter applied to these plots stacks a total of 21 traces andthe data and filter have a trace-to-trace spacing of 6.25 m (the mid-point spacing—half of the receiver spacing). This spatial samplingavoids any aliasing problems and attacks all events not having thedip (slope) of the fault-plane reflection. In particular, it attacks thehorizontal reflections.

To demonstrate the action of the dip filter, we show both migratedimages and their dip-filtered versions in Fig. 6. The panels (a) and(b) of Fig. 6 are for Model 2 and panels (c) and (d) of Fig. 6 are forSlip Model A embedded in Model 1. Panels (a) and (c) of Fig. 6 showthe migrated images and panels (b) and (d) show the images in (a)and (c) after dip filtering. Note that the model with a pore-pressurecontrast, Model 2, reflects waves much like a traditional seismic in-terface in that the reflection coefficient is independent of frequency.In contrast, the model with a slip interface, Slip Model A embeddedin Model 1, has a reflection coefficient which is approximately thederivative of the incident wave (Chaisri & Krebes 2000). A slice

cut out of the dip-filtered images in the direction perpendicular tothe fault-plane (shown as a white arrow in the panels (b) and (d) ofFig. 6) helps in assessing the accuracy of the numerically simulatedfault-plane reflections. In Fig. 7, we plot the reflected waveformstogether with either the incident wavelet (a 20 Hz Ricker wavelet)in panel (a) or the derivative of the incident wavelet in panel (b),depending on whether the model contains !P or interfacial slip atthe fault. Note that these plots are in depth—the incident wavelet(or its derivative) has been plotted in depth in Fig. 7 using the lo-cal wave speed to make the time-to-depth conversion—and that theamplitudes have been normalized. Thus, the agreement seen be-tween the numerical and predicted waveforms in Figs 7(a) and (b)demonstrates that the SEM modelling, processing and dip-filteringtogether produce an accurate shape of the reflected waveform fromthe fault plane. In the next section, we examine reflectivity from thefault plane for combinations of the basic fault models: juxtapositioncontacts, pore-pressure contrasts and models with interfacial slip atthe fault plane embedded in one of the pore-pressure contrast mod-els. The purpose of this modelling exercise is to study the characterof the various types and combinations of reflectivity associated withthe fault plane.

8 A M P L I T U D E S O F WAV E SR E F L E C T E D F RO M D I F F E R E N TFAU LT M O D E L S

Although many possible combinations of the different faulted mod-els with !P and slip interfaces exist, we chose to highlight twogroups of four models in the following which demonstrate the mostpertinent information regarding the nature of fault-plane reflections.The first group comprises the four Models (1–4) with different !Pacross the fault and allows the comparison of how different valuesof !P leave their imprint on the fault-plane reflections. The secondgroup comprises the four models with different degrees of interfacialslip at the fault-plane. For this comparison, each of the Slip-Models(A–D) are embedded in a background model given by Model 2 (seeTables 1 and 2). Recall that Model 2 is the model with the maximumamount of !P at the fault plane. Thus, embedding the Slip-Modelsin Model 2 tests to what degree the different slip interfaces show upin the seismic data relative to the maximum amount of !P acrossthe fault in the employed models.

In Fig. 8, we plot the maximum reflected amplitude within a smalltime window (100 ms) of the fault-plane for the four different mod-els of !P across the fault: 600 psi (Model 2), 300 psi (Model 3),150 psi (Model 4) and 0 psi (Model 1—a juxtaposition model). Notethat there is not a slip interface at the fault in these examples. Allof these amplitude maps are plotted for the cases with (solid line)and without (dashed line) 20 per cent Gaussian additive noise cor-rupting the migrated images. The plots demonstrate that Gaussianadditive noise is efficiently attacked by the dip-filtering step, giv-ing roughly the 1/

(N attenuation of noise exploited in stacking

(Haney et al. 2005a). Comparison of the plots shows a roughly lin-ear relation between the reflection amplitudes and the amount of!P. As expected, the reflectivity due to the juxtapositions has beendampened by the dip-filtering applied in the direction of the fault-plane. For a pore-pressure contrast as small as 150 psi, it is difficultto tell if a pore-pressure contrast exists at all. Specifically, note thesimilarity of Figs 8(c) and (d), especially in the degree with whichthe wobbles due to the juxtapositions contribute to the fault reflec-tivity. Fig. 9 depicts the zero-offset migrated sections from whichthese amplitude maps are made. As such, the degree to which the


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Figure 6. Migrated seismic images are shown in the left-hand panels, (a) and (c) and the same images, after applying dip-filtering to highlight the fault-planereflections, are shown in the right-hand panels, (b) and (d). Panels (a) and (b) are for Model 2—a pore-pressure contrast across the fault (see Table 1). Panels(c) and (d) are for Model 1—no pore-pressure contrast across the fault (see Table 1). Although there is no pore-pressure contrast in panels (c) and (d), the faultplane is a slip interface. The parameters describing the slip interface are referred to as Slip model A (see Table 3) and, hence, we refer to the model used inpanels (c) and (d) as Slip model A embedded in Model 1. The traces in Fig. 7 are sliced from these images in a direction normal to the fault-plane, as shownby an arrow in the dip-filtered images. The horizontal events in the upper panels appear to be not as well suppressed as in the lower panels simply because thefault-plane reflection is stronger in the bottom panel and, as a result, the maximum amplitude of the display is higher.

fault-plane reflections show up in the zero-offset migrated sectionsprior to dip-filtering in the direction of the fault-plane can be as-sessed.

In Fig. 10, we again plot the maximum reflected amplitude withina small time window (100 ms) of the fault-plane for Slip-Models A,B, C and D. As discussed before, these Slip-Models are embedded ina background model given by Model 2. From Fig. 10, it is seen thatthe amplitude of the fault-plane reflection attributable to interfacialslip is reduced as the normal and shear compliances are decreased ingoing from Slip-Model A to D. The reduction in reflected amplitudeis roughly linear in proportion to the reduction of the compliances.This occurs because the reflection coefficient is proportional to thecompliance for a relatively weakly slipping interface, as shown ineq. (A6) of Appendix A.

The linearization employed in the derivation of eq. (A6) in Ap-pendix A is valid for the Slip-Models considered here because thedimensionless shear and normal compliances of the interfaces are

considerably less than unity. The dimensionless compliance, andnot compliance itself, is the measure of how strongly reflecting aslip interface is, since the dimensionless compliance accounts fordifferent frequencies of the incident wave and different local mate-rial properties. From Appendix A, we know that the dimensionlessnormal compliance is )'N $* sec + , where ) is the dominant fre-quency of the incident wave (20 Hz), 'N is the normal compliance,$ is the average density across the interface, * is the average P-wavevelocity across the interface and + is the incidence angle. For nor-mal incidence (+ = 0&), an average density $ = 2260 kg m#3

and an average P-wave velocity v p = 2700 m s#1, the dimen-sionless normal compliance varies between 0.38 and 0.08 for Slip-Models A–D. When this dimensionless number is less than unity,we may expect the strength of the reflections to be linear as a func-tion of the compliance, as observed in Fig. 10 for the Slip-Models.Similarly for tangential case, the dimensionless tangential compli-ance is )'T $% cos + . For normal incidence (+ = 0&), an average



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Figure 8. Amplitude along fault-plane for Model 2 in panel (a), Model 3 in panel (b), Model 4 in panel (c) and Model 1 in panel (d). See Tables 1 and 2 fora description of the different Models. These amplitude maps are plotted both with (solid line) and without (dashed line) 20 per cent Gaussian additive noise.The amplitude is plotted as a function of depth on the fault-plane and the extent of the fault-plane is shown with a horizontal line in the bottom portion ofeach plot. These models represent the situations of: 600 psi pore-pressure difference across the fault (a), 300 psi pore-pressure difference across the fault (b),150 psi pore-pressure difference across the fault (c) and 0 psi pore-pressure difference across the fault; in other words, the juxtaposition contact model (d).There is not a slip interface at the fault in these examples.

density $ = 2260 kg m#3 and an average S-wave velocity v p =1200 m s#1, the dimensionless tangential compliance varies between0.34 and 0.07 for Slip-Models A–D. Thus, it is entirely expectedthat the reflection amplitudes scale linearly with compliances inFig. 10.

For the smallest compliance, Slip-Model D, the magnitude of thereflection is on the same order as that of the reflection caused by the!P present in Model 2 (600 psi). Such a similarity is seen by com-paring Figs 8(a) and 10(d). Thus, the degree of slip in Slip-ModelD would seem to be a lower limit of compliance for indications of

interfacial slip to appear when !P = 600 psi across the fault. Fig. 11depicts the zero-offset migrated sections from which the amplitudemaps in Fig. 10 are made. The reflections due to the interfacial slip atthe fault-plane can clearly be made out even without the dip-filteringstep.

As demonstrated by Fig. 8, a linear relation may be assumed be-tween the magnitude of the fault-plane reflection due to a !P andthe value for !P itself. This linear behaviour is due to the fact thatthe values of !P considered here do not drastically modify the elas-tic parameters across the fault. That is, the changes in the elastic


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Figure 9. Zero-offset migrated sections for Model 2 in panel (a), Model 3 in panel (b), Model 4 in panel (c) and Model 1 in panel (d). See Tables 1 and 2 for adescription of the different Models. The amplitude plots along the fault plane in Fig. 8 are derived from these zero-offset sections after dip-filtering.

parameters across the interfaces divided by their average valuesacross the interface, !*/*, !%/% and !$/$, are less than unity.Since a linear relation also exists between the degree of interfacialslip and the strength of reflection, we can estimate the minimal de-gree of slip for a reflection from a slip interface to show up in thepresence of juxtapositions only (when !P = 0 psi). Given that Slip-Model D is the minimal amount of slip for notable reflections in thepresence of a !P = 600 psi (Model 2), a slip interface with shear andnormal compliances reduced by one-fourth of those in Slip-Model 4would constitute the threshold for indications of interfacial slip toappear in Model 4, when !P = 150 psi across the fault. As alsodemonstrated by Fig. 8, fault reflectivity from pure juxtapositions(Model 1, !P = 0 psi) are roughly half as strong as in the case of!P = 150 psi. Thus, a slip interface with shear and normal com-pliances reduced by one-eighth of those in Slip-Model 4 wouldconstitute the threshold for indications of interfacial slip to appearwhen !P = 0 psi across the fault. From Table 3, such a mini-mally (seismically) visible slip interface would have 'N = 1.25 %10#11 m Pa#1 and 'T = 2.5 % 10#11 m Pa#1. Reflections from indi-vidual slip interfaces with compliances less than these values wouldbe impossible to make out in the presence of the reflectivity due tojuxtapositions at the fault-plane, given the typical elastic propertiesfor sands and shales at South Eugene Island field.

9 R E L AT I N G S L I P I N T E R FA C E S T O AP R E S S U R I Z E D FAU LT

The slip interfaces used in the numerical modelling and shown inTable 3 do not, up to this point, have any physical meaning in termsof the pore-pressure locally at the fault. In this section, we relatea slip interface to an effective-layer model that demonstrates muchof the same wave-scattering behaviour. Thus, a single slip interfacecan thought to model a ‘family’ of thin layers with varying elasticproperties and thicknesses. With an effective thin layer describedin terms of its thickness, density and velocity, the empirical rela-tionships between effective stress and density and effective stressand velocity we derived earlier can lend the slip interfaces physi-cal meaning in terms of pore-pressure. Note that, in what follows,the relationships we derive for compliance of an effective thin layershould not be interpreted physically in terms of an actual fracture.What we are doing here is simply relating a slip interface to a familyof thin layers in terms of the similarity of their reflected waveforms.

The derivation presented here is for normally incident P waves;we focus on normally incident P waves since the seismic imagingdiscussed in previous sections utilized PP-scattered waves at smallincidence angles. In fact, for a given spreadlength of a seismic sur-vey, the incidence angles for a fault plane reflection are smaller than



800 900 1000 1100 1200 1300

0

0.04

0.08

0.12

0.16

depth (m)am

plitu

de800 900 1000 1100 1200 1300

0

0.04

0.08

0.12

depth (m)

ampl

itude

800 900 1000 1100 1200 1300

0

0.03

0.06

0.09

depth (m)

amp

litu

de

800 900 1000 1100 1200 1300

0

0.02

0.04

depth (m)

amp

litu

de

(a) (b)

(c) (d)

Figure 10. Amplitude along fault-plane for Slip-Model A in panel (a), Slip-Model B in panel (b), Slip-Model C in panel (c) and Slip-Model D in panel (d).See Table 4 for a description of the different Slip-Models. All Slip-Models shown here are for a fault embedded in Model 2 (see Tables 1 and 2) having apore-pressure contrast of 600 psi across the fault. These amplitude maps are plotted both with (solid line) and without (dashed line) 20 per cent Gaussianadditive noise. The amplitude is plotted as a function of depth on the fault-plane and the extent of the fault-plane is shown with a long horizontal line in thebottom portion of each plot. The subportion of the fault where interfacial slip occurs is shown beneath this line with a shorter horizontal line. The maximumamplitude occurs near the centre of the slipping portion of the fault in each slip-model. The amplitude map resembles a triangle since it is, roughly speaking,the convolution of two boxcar functions: the slipping portion of the fault-plane and the dip filter. In moving from Slip-Model A to D, the compliance of thefault slip becomes smaller and, as a result, the reflection magnitude scales in the same proportion, as predicted by eq. (A6) in Appendix A for a weakly slippinginterface. The different Slip-Models invoke slip at the fault to represent weakness due to elevated pore-pressure.

those for a horizontal interface at the same depth. Our effective-layermodel begins from a weak scattering assumption. For a thin layer, ifthe interface reflection coefficients at the upper and lower boundaryare small, then the entire series of reverberations (Aki & Richards1980) within the layer can be neglected. The total reflection coef-ficient from the thin layer can thus be approximated simply as thesum of the reflections off the upper and lower interface. For the casewhen the thin layer is sandwiched between two identical media,

Rtot ) RPP # RPP exp"

2i)h*L

#

, (14)

where RPP is the P wave to P wave (PP) reflection coefficient at theupper interface (the reflection at the lower interface is # RPP), )

is the frequency, h is the thickness of the thin layer, and * L is theP-wave velocity in the thin layer. In eq. (14), we have assumed thatthe impedance difference between the thin layer and the host mediumis small enough that the transmission coefficients in moving fromthe host medium into the thin layer and vice versa are close to 1.This is consistent with the weak-scattering assumption.

The next approximation relies on the layer being sufficiently thin.If, for the argument of the exponential term in eq. (14),

2)h*L

* 1, (15)

then the exponential can be expanded to first order in a Taylor series

exp"

2i)h*L

#

) 1 + 2i)h*L

. (16)

Note that the condition in eq. (15) states that 1 + 4,h/(L , where (L

is the wavelength of the wave in the thin layer. Hence, the condition

means that only a fraction of a wavelength fits in the layer. Insertingthe Taylor series approximation into eq. (14) gives

Rtot ) #RPP2i)h*L

. (17)

This expression shows that the total reflection from a thin weak layeris proportional to the derivative of the incident wave. Widess hasdiscussed this fact in a paper on vertical seismic resolution (Widess1973).

From eq. (A6) in the Appendix, the normal incidence PP reflec-tion coefficient for a weakly slipping interface between two identicalmedia (the host medium) is

RsP P ) i)'N $*

2, (18)

where ) is the frequency, 'N is the normal compliance, $ is thedensity of the host medium, * is the P-wave velocity of the hostmedium, and the superscript s indicates that this is the reflectioncoefficient for a slipping interface. This equation comes with itsown assumption, namely that the dimensionless normal complianceis much smaller than 1, )'N $* * 1. Equating this expression toeq. (17) gives

i)'N $*

2= #RPP

2i)h*L

. (19)

By canceling common factors and solving this for 'N , the normalcompliance, we get

'N = # 4h$**L

RPP . (20)


946 M. Haney et al.

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2.0 2.5 3.0 3.5 4.0x105

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x105

2.0 2.5 3.0 3.5 4.0


depth(km)

(a) (b)

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2.0 2.5 3.0 3.5 4.0x105

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2.0 2.5 3.0 3.5 4.0


depth(km)

(c) (d)

Figure 11. Zero-offset migrated sections for Slip-Model A in panel (a), Slip-Model B in panel (b), Slip-Model C in panel (c) and Slip-Model D in panel(d). See Table 4 for a description of the different Slip-Models. All Slip-Models shown here are for a fault embedded in Model 2 (see Tables 1 and 2) havinga pore-pressure contrast of 600 psi across the fault. The amplitude plots along the fault plane in Fig. 10 are derived from these zero-offset sections afterdip-filtering.

In the weak scattering approximation, we can substitute a weak-scattering approximation for the normal-incidence welded-interfacereflection coefficient RPP. This can be obtained from eq. (A6) in theAppendix by ignoring the terms related to slip

RPP = 12

"

!$

$+ !*

*

#

, (21)

where !$ = $L # $, $ = ($L + $)/2, !* = *L # * and * =(*L +*)/2. Rewriting eq. (21) in terms of the properties of the layerand host medium gives

RPP = $L # $

$L + $+ *L # *

*L + *. (22)

Substituting this expression into eq. (20) for RPP yields

'N = 4h$**L

"

$ # $L

$ + $L+ * # *L

* + *L

#

. (23)

Since 'N is by definition greater than zero, the effective thin layer isallowed to have relatively lower density $ > $ L and lower velocity* >* L than the host medium. This is the case of a locally pressurizedfault, since velocity and density decrease with increase in pore-pressure. We note that the weak-scattering approximation employed

here should be appropriate for PP reflection from an overpressurizedfault since, according to the pore-pressure relationships describedearlier, *L , 0 does not occur for any value of the pore-pressure.This may not be the case for shear waves, though, since the shearwave velocity can go to nearly zero at zero effective stress (Zimmeret al. 2002).

Using the effective–stress relationships derived in previous sec-tions for $ L and * L in the case of unloading, $u

L (" d ) and *uL (" d ),

and fixing the depth (or, equivalently the lithostatic stress " v) sothat the effective stress varies only with pore-pressure (" d = " v #p), the compliance of a fracture can be put in terms of the thicknessof the layer and the pore-pressure

'N (h, P) = 4h$**u

L (P)

%

$ # $uL (P)

$ + $uL (P)

+ * # *uL (P)

* + *uL (P)

&

. (24)

Note that we use the unloading relationships for $ L and * L . Thisis because a slip interface is indicative of a locally weak fault. Therelative weakness is due to the fault being overpressured, where byoverpressured we mean its effective elastic properties are modifiedby pore-pressure as seen in the well log data of the previous sec-tions. The overpressure in a fault could perhaps be from pressurized



fluid migrating up the fault plane, as recently observed by Haneyet al. (2005b). Hence, the correct pressure curve to use is the un-loading curve, since the migrating fluid is entering the pore spaceand unloading the fault zone sediments. This detail has also beendiscussed by Revil & Cathles (2002). The unloading curve requiresan additional parameter, the maximum past effective stress " max thatthe fault rock has experienced before being unloaded to its currentstate. With this information, the slip models used in the previousmodelling section can be related to an effective thin layer (or faultzone) described by four parameters: a fault zone of thickness h, fixeddepth z, excess fluid pressure P # P h and maximum past effectivestress " max.

Based on these considerations, the value of the pore-pressurecorresponding to a particular linear-slip interface depends on threeother parameters besides the pore-pressure. In Table 4, we fix twoof these parameters—the depth of the fault zone (1850 m) and thethickness of the fault zone (10 m). We take the thickness of the faultbased on the well logs that penetrated the B-fault zone. The tableshows the values of the effective thin layer for each Slip-Model,A–D, while " max varies between 1600 and 2800 psi. The table alsoshows the effective stress corresponding to those values of velocityand density along the unloading path, and uses the assumed depthto convert the effective stress to pore-pressure. We do not show re-sults for variation with thickness h since its effect is a simple linearscaling, as seen in eq. (23). The variation with depth z is also fairlyunimportant since it changes only the value of the pore-pressure fora given effective stress. Note that, in Table 4, certain values of thecompliance do not exist for some values of " max since the compli-ance does not fall in the range of possible compliances (i.e. it isnot positive) based on the values of density and velocity. Hence,some small compliance values cannot be modelled for certain com-binations of h, z and " max. In order to obtain smaller compliancevalues in these cases, the thickness of the fault would need to bemade smaller because the compliance, from eq. (23), scales linearlywith h.

In the A10ST well at South Eugene Island, the effective stress inthe B-fault zone is measured to be as low as 166 psi (Losh et al.1999), much lower than the effective stresses in the adjacent downor upthrown sediments. In other words, the fault itself is locallyoverpressured and weak compared to the host rock. Therefore, someof the low values for effective stress shown in Table 4, while unusual,are entirely possible for the growth faults at South Eugene Island.Anomalously low effective stresses of 575 and 807 psi were alsomeasured in the A20ST well as it crossed the A-fault system (Loshet al. 1999). The locally high pore-pressures in the B-fault zonereported in Losh et al. (1999) have been implicated as the cause ofanomalously high reflectivity observed at the B-fault plane (Haneyet al. 2005b). Based on our numerical modelling, a plausible modelfor the B-fault zone in the location of the reported anomalously highreflectivity would be either Slip-Model A or B as shown in Table 4,for the case of " max = 2000 psi.

1 0 C O N C L U S I O N

We have presented a complete numerical modelling experiment byutilizing an SEM implementation of the 2-D elastic wave equa-tion and processing the resulting waveforms into their time-migratedimages. We derived a simple dip filter and used it to isolate fault-plane reflections. We then exploited the relationships between theelastic parameters, density and velocity, to create physically mean-ingful models of sealing faults that maintain a !P of up to 600 psi.For these !P models, we assumed that the overpressure mechanism

is purely due to undercompaction. In the course of this modelling,we found that the minimum !P necessary to give rise to substantialfault-plane reflections is on the order of 150 psi at the South Eu-gene Island field. We have found evidence in field data arising fromthe lack of a reflection from the F-fault (see Fig. 1) supporting thisestimate (Haney, 2005).

Taking advantage of the SEM modelling code’s ability to accom-modate linear-slip interfaces, we selected four different values ofthe normal and shear compliances for the fault interface. We foundthat the reflections from the slip interfaces dominate the reflectionsfrom pore-pressure contrasts across the fault for compliance valuesabove $10#10 m Pa#1. By noting that the different slip interfaceswere in the weak scattering regime, we estimated the minimal val-ues of the compliances necessary to produce a notable reflection at afault without any !P across it. Looking for physical insight into themeaning of the slip interfaces, we derived, from a weak-scatteringmodel, an equivalent thin, weak layer that gives virtually the samereflection as a linear-slip interface. We used this equivalent layermodel to relate the slip interface to realistic values of pore-pressurein a fault zone at South Eugene Island. To do so required exten-sive use of the effective–stress relationships for the unloading pathsderived from well logs at the South Eugene Island field.

Interpreting fault zone properties from fault-plane reflections isthus closely tied to knowledge of the subsurface distribution offluids (Hatchell 2000) and stress histories. By paying attention tothe details of the rock physics of pore-pressure and utilizing ad-vanced numerical techniques for modelling wave propagation, thephysical mechanisms giving rise to fault-plane reflections can bethoroughly investigated. Understanding the relative importance ofthese physical mechanisms makes possible the detection of faultproperties through the prevalent techniques of seismic reflectionimaging.

A C K N O W L E D G M E N T S

Thanks to Jon Sheiman from Shell International Exploration andProduction for providing valuable input on rock physics and frac-tures. Shell International Exploration and Production funded thisresearch under the Gamechanger program. Fig. 1 was constructedusing the Generic Mapping Tool (GMT) version 4.0.

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A P P E N D I X A : S P E C T R A L E L E M E N TM O D E L L I N G O F A L I N E A R - S L I PI N T E R FA C E

We chose the SEM to simulate fault reflectivity for its abilityto allow a free-form mesh and in order to include the possibil-ity of slip at interfaces in our numerical models. As evidence ofSEM’s ability to handle challenging boundary conditions, it has re-cently been applied to wave propagation near a fluid–solid interface(Komatitsch et al. 2000). Interfacial slip had been implementedpreviously in SEM2DPACK, the 2-D SEM code available onlineat the Orfeus Seismological Software Library: http://www.orfeus-eu.org/links/softwarelib.htm. SEM2DPACK was developed by oneof the authors, J.-P. Ampuero, for the simulation of earthquake dy-namics (Ampuero 2002). In this work, we adapted the interfacecondition to a linear-slip law.

For incident P–SV waves, the linear-slip law is a boundary con-dition expressed as (Schoenberg 1980)

t · n = Z#1!u (A1)

!u =!

u+z # u#

z

u+x # u#

x

$

, Z#1 =!

'#1T 0

0 '#1N

$

,

t =!

"xx "xz

"xz "zz

$

, n =!

0

1

$

, (A2)

where the superscript (#) refers to the side of the interface on whichthe wave is incident, (+) the other side of the interface, u x and u z

are the horizontal and vertical particle displacements, " xx and " zz

are the normal stresses and " xz is the shear stress. The particularchoice of the normal vector in eq. (A2) means that we consider herea horizontal slip interface. The SEM code has no such restriction;any orientation of a slip interface can be handled. The matrix Z#1

appearing in eqs (A1) and (A2) is the stiffness matrix of the slipinterface (inverse of the compliance matrix). The fact that Z#1 isdiagonal means we only consider rotationally symmetric slip inter-faces (Haugen & Schoenberg 2000) in our SEM implementation.The parameters 'N and 'T are the normal and tangential compli-ances, respectively, and these quantify the degree of slip along theinterface. For 'N = 0 and 'T = 0, the interface is welded, and for'N = - and 'T = -, it is a stress free surface.

The boundary condition described by eqs (A1) and (A2) can beobtained in the limit of a thin, weak layer in welded contact with

its surrounding rock. As a result, linear-slip has been suggestedas a good model for scattering from faults and fractures (Coates& Schoenberg 1995). With this in mind, it is important not toconfuse the slip model in eqs (A1) and (A2) with slip that occursalong a fault during an earthquake. The linear-slip model entailssome slipping at the interface that is the order of particle displace-ments during the passage of a seismic wave ($10#6 m). Active,earthquake-generating faults typically slip on a length scale threeto four orders of magnitude larger ($10#3–10#2 m). Earthquakeslip is also hysteretic, whereas interfaces undergoing linear-slip re-turn to their equilibrium state after the seismic wave has movedon.

To implement the linear-slip model in SEM, the weak form of theequation of motion is needed (Komatitsch & Tromp 1999)'

-

$w · .2t u d3x = #

'

-

.w: t d3x +'

/

[t · n] · w d2x, (A3)

where u is the displacement, $ is the density, t is the stress tensor,t·n is the traction on the slip interface, and w is the test function. Thesemi-colons in eq. (A3) represent tensor contraction. The last termon the right-hand side of eq. (A3) is the contribution from the slipinterface /, which is here taken to be planar. At the slip interface,we employ split nodes (Andrews 1999) in the spectral element meshto accurately model a fracture.

After discretizing the displacement field and assembling theglobal mass and stiffness matrices, eq. (A3) can be written as thematrix equation (Komatitsch & Tromp 1999)

MU = #KU + BT, (A4)

where M and K the mass and stiffness matrices, respectively, U isthe displacement vector of the global system, and T is the tractionvector of the global system. The last term is non-zero only on thepart of the boundary where slip occurs; this restriction is imposedby the matrix B. The essence of this SEM implementation of aslip interface is that the two separate meshes on either side of theslip interface (denoted here as mesh 1 and mesh 2) are put intocommunication via the last term in eq. (A4) by using the slip law ineq. (A1)

M1U1 = #K1U1 + '#1N B1!U z

1 + '#1T B1!U x

1

M2U2 = #K2U2 # '#1N B2!U z

2 # '#1T B2!U x

2 , (A5)

where the asymmetry of the ± signs between the two last termsis in accordance with Newton’s third law. The subscripts 1 and 2indicate to which mesh the variables belong. The superscripts z andx label the normal and tangential component of the displacementdiscontinuity. In the formulation we have outlined here, the slip law,eq. (A1), enters into the equation of motion by a substitution of theslip for the traction at the fault. Note that, due to the complianceof the slip interface appearing in the denominator of eq. (A5), thenumerical scheme experiences conditional instability for too smalla compliance.

The SEM implementation of eq. (A5) utilizes an explicit New-mark scheme (Zienkiewicz & Taylor 2000) consisting of a predictor,a solver and a corrector step (Komatitsch & Vilotte 1998) as shownin Table (A1). Waveforms computed with this implementation aredisplayed in Fig. A1. Our model for this example is an elongated2-D block, whose side boundaries, shown as dashed in Fig. A1,are periodic and whose upper and lower boundaries are absorbing.Consistent with the periodic boundary condition, we excite a uni-directional plane P wave near the top of the block. In front of theplane wave, we measure the wavefield with ten receivers. A slip


950 M. Haney et al.

Table A1. The predictor–corrector algorithm: n stands for the time step, U is the acceleration, !tis the time step increment, U is the displacement and V is the velocity.

for n = 1 : N

PredictorU 1(n+1) = U 1(n) + 1

2 !t V 1(n)U 2(n+1) = U 2(n) + 1

2 !t V 2(n)V 1(n+1) = V 1(n)V 2(n+1) = V 2(n)

SolverSolve eq. (A5) for both U1(n+1) and U2(n+1) usingthe predicted values U 1(n+1), U 2(n+1), V 1(n+1) and V 2(n+1).This is straightforward since M 1 and M 2 are diagonal.

CorrectorU 1(n+1) = U 1(n+1) + 1

2 !t V 1(n) + 12 !t2 U1(n+1)

U 2(n+1) = U 2(n+1) + 12 !t V 2(n) + 1

2 !t2 U2(n+1)V 1(n+1) = V 1(n+1) + !t U1(n+1)V 2(n+1) = V 2(n+1) + !t U2(n+1)

end

5

89

76

432

2000 m

200 m

1

10

5 10 15 20 250

0.2

0.4

0.6

0.8

1

frequency (Hz)

refle

ctio

n co

effic

ient

0 0.2 0.4 0.6 0.8 1 1.2600

400

200

0

200

400

600

time (s)

dist

ance

from

inte

rfac

e (m

)

(a)

(b)

(c)

Figure A1. Numerical simulation of a normally incident P wave scattering from a linear-slip interface. In panel (a) is the model, showing ten numbered receiverlocations and a plane wave incident from the upper end of the model. The linear-slip interface is between the receivers 5 and 6. In panel (b) are seismograms takenat each of the receivers. At t = 0.6 s, the incident wave reflects from the linear-slip interface. The media above and below the linear-slip interface are identicaland, therefore, no reflection would occur had there been no slip. In panel (c) is a quantitative comparison between the reflection coefficient calculated from theuppermost seismogram, receiver 10 (circles), and the theoretically exact reflection coefficient for this slip interface (solid line). The reflection coefficient forreceiver 10 is calculated by isolating the incident and reflected waveforms and taking their spectral ratio. The agreement validates the numerical technique usedto model slip interfaces. Also shown as a dashed line is the approximate reflection coefficient given by eq. (A6). The approximation becomes progressivelyworse for higher frequencies.

interface cuts through the center of the block, between receivers5 and 6, which is characterized by a normal compliance of 2.2 %10#9 m Pa#1. The slip interface has also a shear compliance, butthe P wave is incident normally and, therefore, excites no shear. The

media on either side of the slip interface are identical with density,P- and S-wave velocity of 2300 kg m#3, 2000 m s#1 and 1000 m s#1,respectively. The plane wave source waveform is a Ricker waveletwith a dominant frequency of 10 Hz.



Since Chaisri & Krebes (2000) have previously solved for thefrequency-dependent reflection coefficient from a slip interface, wecan check our numerical result against the analytic solution. In thecase of weak elastic contrasts across the slip interface, small (di-mensionless) compliances, and small angles of incidence, the lin-earized PP-reflection coefficient at a planar slip interface is a goodapproximation to the true reflection coefficient. The PP-reflectioncoefficient in this case is given by:

RPP )!

12

# 2"

%

*

#2

sin2 +

$

!$

$+ 1

2s2+

!*

*

# 4"

%

*

#2

sin2 +!%

%+ i

!

12

# 2"

%

*

#2

sin2 +

$

s+)'N $*

#2i"

%

*

#3

cos + sin2 +)'T $%, (A6)

where !*, !% and !$ are the changes in P-, S-wave velocity anddensity of the two half-spaces, *, % and $ are the average P-, S-wavevelocity and density of the two half-spaces, ) is the angular fre-quency of the incident wave, 'T and 'N are the tangential and normal

compliances of the interface, and + is the reflection/incidence angle.As previously stated, the approximation in eq. (A6) holds for smallrelative changes in the medium parameters (!*/*, !%/%, !$/$ *1), small dimensionless shear compliance ()' I $% cos + * 1), smalldimensionless normal compliance ()'T $* sec + * 1), and smallangles of incidence. Note that, for the case of small dimensionlesscompliances, the reflected wave due to a pure slip interface betweentwo identical half-spaces is proportional to the derivative of the in-cident wave.

We plot the wavefield interacting with the slip interface at eachof the 10 receivers in Fig. A1. For the value of slip used in thisexample (2.2 % 10#9 m Pa#1), the dimensionless normal complianceis relatively small and the reflection from the slip interface is wellapproximated by eq. (A6). We show a quantitative benchmarkingbetween the numerically calculated reflection coefficient (shownby circles) and the exact theoretical reflection coefficient (depictedby the solid line) in the lower panel of Fig. A1. The agreementbetween the two curves supports the numerical implementation forslip interfaces used in this paper. The approximate expression forthe reflection coefficient, eq. (A6), is shown in Fig. A1 as a dashedline and is seen to become a worse approximation with increasingfrequency.


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